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Tuesday, December 13, 2016

You can find the secret to doing mathematics in a tubeless bicycle tire

The author climbing the locally-notorious Country View Road just south of
San Jose, CA

As regular readers may know, one of my consuming passions in life besides
mathematics is cycling. Living in California, where serious winters were wisely banned
many years ago, on any weekend throughout the year you are likely to find me out on a
road- or a mountain bike.

Being also a lover of well-designed technology, I long ago switched to using tubeless
tires on my road bike. Actually, it’s bikes, in the plural—my road bikes number four, all
with different riding conditions in mind, but all having in common the same kind of ultra-
narrow saddle that non-cyclists think must be excruciatingly painful, but is in fact
engineered to be the only thing comfortable enough to sit on for many hours at a
stretch. [Keep going; I am working my way to making a mathematical point. In fact, I am
heading towards THE most significant mathematical point of all: What is the secret to
doing math?]

Road tubeless tires have several advantages over the more common type of tire, which
requires an airtight innertube. One advantage is that you need inflate them only to 80
pounds per square inch, as opposed to the 110 psi or more for a tubed tire, which
provides even more comfort over those many hours in the saddle.

You need tire pressures 3 or more times that of a car tire because of the extremely low
volume in a road-bike tire, which sits on a 700 cm diameter wheel with a rim whose
width is between 21 mm and 25 mm. It is that high pressure that made the manufacture
of tubeless wheels and tires for bicycles such a significant challenge. How can you
ensure an almost totally airtight fit when the tire is inflated, and it still be possible for an
average person to remove and mount a deflated tire with their bare hands. (Tire levers
can easily damage tubeless wheels and tires.) We are almost to the secret to doing
math. Hang in there.

Clever design: Tubeless rims and tires on a road bike wheel.

The airtight fit is possible precisely because of that relatively high pressure inside the
tire—80 psi is over five times the air pressure outside the tire. (An automobile tire is
inflated to roughly twice atmospheric pressure, much lower.)
The cross-sectional photo on the left shows how a tubeless tire has a squared-off ridge
that fits into a matching notch in the rim. The more air pressure there is in the tire, the
tighter that ridge binds to the rim, increasing the air seal.

The problem is, as I mentioned, getting the tire on and off the rim. The tire ridge that fits
into the rim-notch has a steel wire running through it, and its squared-off shape is
designed to make it difficult for the tire to separate from the rim—that, after all, is the
point. To solve the mounting/removal problem, the wheel has a channel in the middle,
as shown more clearly in the photo on the right.

To mount the tire, you push the two tire-rims into that channel, one after the other. By
the formula for the circumference of a circle, when a tire rim is in that center channel,
you have just over 3 times the depth of the channel of superfluous tire length to play
with, roughly 12mm of tire looseness. The idea is to use that “looseness” to work your
way around the wheel, pushing (actually rolling) first one tire edge over the wheel rim
and into the channel, then the other. Once the tire is seated on the rim, inflating it with a
hand pump forces the tire rims out of the channel into the notches. To remove the tire
after it is deflated, you push the two tire rims into the channel and reverse the process.

That, at least, is the theory. Putting theory into practice turns out to be quite a challenge.
When I first started to use road tubeless tires, several years ago, I read several online
manuals and watched a number of YouTube videos demonstrating how to do it, and
could never do it. I usually ended up taking the wheel and tire to my local bike shop,
where the mechanic would do it for me with seeming ease before my eyes. “Fifteen
dollars, please.”

But what would happen if I had a flat on one of the remote rides I regularly do in the
mountains that surround Silicon Valley, where I live? One major advantage of tubeless
tires is that, even if they puncture, usually the air leaks out only very slowly, and can
generally be stopped by inflating the tire from a small pressure-can of air and liquid latex
you carry in your back pocket, which seals the hole. Which is how I was always able to
get to a bike shop where someone else could solve the problem for me. But a major
puncture in the remote, with no cell phone access, could leave me dangerously
stranded. Clearly, I had to learn how to do it myself.

From now on, when I say “change a tubeless tire”, you can interpret it as “do
mathematics”. The secret is coming up. Moreover, it is coming with a moral that those of
us in mathematics education ignore at our students’ peril.

What I find cool is that, for me I somehow stumbled on the secret to doing math fairly
early in life, before math had become such a problem that I felt I could never do it. But
taking up cycling later in life, when I had a fully developed set of metacognitive skills, I
approached the problem of changing a tubeless tire in much the same way as many
people—including, I suspect, the mechanics in my local bicycle shop—see math.
Namely, people like me (and that smart kid sitting in the front row in the school math
class) make doing math look effortless, but many people feel they could
never master it in a million years.

Nothing, surely, can look less requiring of skill or expertise than putting a tire on a
bicycle wheel. (This is why I think this is such a great example.) Surely, you just need to
read an instruction manual, or perhaps have someone demonstrate to you. But no
matter how many times I read the instructions, no matter how many times I
viewed—and re-viewed—those how-to YouTube videos, and no matter how many times
I stood alongside the bike shop mechanic and watched as he quickly and effortlessly
put the tire onto the wheel, I could never do it.

Just think about that for a moment. For some tasks, instruction (on its own) just does
not work. Not even for the seemingly simple task of changing a bicycle tire. And yet we
think that forcing kids to sit in the math class while we force-feed instruction will result
in their being able to do math! Dream on.

What does work, in fact what is absolutely necessary, both for changing tubeless tires
and doing math, is that the learner has to learn to see things the way the expert does.
And, since instruction does not work, that key step has to be made by the learner. All
that a good teacher can do, then, is find a way to help the learner make that key leap.
[That short initial word “all” belies the human expertise required to do this.]

Clearly, when I was, yet again, standing in the bicycle repair shop, watching the
mechanic change my tire, what he was doing—more precisely, what he was
experiencing—was very different from what I was doing and experiencing when I tried
and failed. What was I not getting?

My big breakthrough finally came the one time when the mechanic, holding
the wheel horizontally pressed to his stomach, while manipulating the tire with both
hands, told me what he was really doing. “You have to think of the tire as alive,” he
said. “It wants to be sitting firmly on the rim” [that, after all, is what it
was—expensively—designed for], “but it is not very disciplined. It’s like a small child. It
moves around and resists your attempts to force it. You have to understand it, and be
aware, through your hands, of what it is doing. Work with it—be constantly aware of
what it is trying to do—so you both get what you want: the tire gets onto the wheel,
where it belongs, and you can inflate it and get back on your bike (where you want to
be).”

Fanciful? Maybe. But it worked. And it continues to work. As a result, not only can I now
change my tubeless tires, it has for me become “mindless and automatic,” as effortless
(to me) as Picasso drawing a simple doodle on a restaurant napkin to pay the bill for his
meal was to him. (I thought that if you got this far, you deserved a second example with
greater cultural overtones.)

It took many years for Picasso to learn to draw the way he did (and for the marketplace
to assign high value to his work), but that does not mean his work was not skillful;
rather, he simply routinized part of it. When I watch a film of him at work, I see
superficially how he created, and it looks routine and effortless, but I do not see his
canvas as he did, and I could not draw as he did.

Likewise, my skill in fitting a tubeless tire, now effortless and automatic, is a result of my
now seeing and understanding what earlier had been opaque.

I admit that it is far easier to learn to mount a tubeless tire on a road bike wheel than to
draw like Picasso. But I am less sure the difference is so great between changing a
tubeless tire and being able to solve any one particular kind of math problem. Still, no
matter how great the difference in the degree of skill required, it is possible to learn from
the analogy.

Given what I have said here, will reading this essay mean you can go out and
immediately be able to change a tubeless tire? Have I just made a case for instruction
working after all? It’s possible—for changing bicycle tires, but surely not for painting like
Picasso. Instruction can and does work, and it is an important part of learning. But my
guess is you would find my words are not enough. I think that the reason that one piece
of bike-shop instruction was so instantly transformative for me was that I had spent an
aggregate of many hours struggling to change my tire and failing. I had reached a stage
where the effective key was to get inside the mind of an expert. But the ground had
to be prepared for that simple revelation to work.

In education, as in so many parts of life, there are no silver bullets. But given enough of
the right preparation—enough experience acquired through repeated trying and
failing—an ordinary lead bullet will do the job.

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The Mathematical Association of America is the world's largest community of mathematicians, students, and enthusiasts. We accelerate the understanding of our world through mathematics, because mathematics drives society and shapes our lives. Visit us at maa.org.